Structural Synthesis of Articulated Manipulators with Non-Fractionated or Fractionated Kinematic Chains without Isomorphism

As the kinematic structure of an articulated manipulator affects the characteristics of its motion, rigidity, vibration, and force transmissibility, finding the most suitable kinematic structure for the desired task is important in the conceptual design phase. This paper proposes a systematic method for generating non-isomorphic graphs of articulated manipulators that consist of a fixed base, an end-effector, and a two-degree-of-freedom (DOF) intermediate kinematic chain connecting the two. Based on the analysis of the structural characteristics of articulated manipulators, the conditions that must be satisfied for manipulators to have a desired DOF is identified. Then, isomorphism-free graph generation methods are proposed based on the concepts of the symmetry of a graph, and the number of graphs generated are determined. As a result, 969 graphs of articulated manipulators that have two-DOF non-fractionated intermediate kinematic chains and 33,438 graphs with two-DOF fractionated intermediate kinematic chains are generated, including practical articulated manipulators widely used in industry.

Graph Deformer Network

Convolution learning on graphs draws increasing attention recently due to its potential applications to a large amount of irregular data. Most graph convolution methods leverage the plain summation/average aggregation to avoid the discrepancy of responses from isomorphic graphs. However, such an extreme collapsing way would result in a structural loss and signal entanglement of nodes, which further cause the degradation of the learning ability. In this paper, we propose a simple yet effective Graph Deformer Network (GDN) to fulfill anisotropic convolution filtering on graphs, analogous to the standard convolution operation on images. Local neighborhood subgraphs (acting like receptive fields) with different structures are deformed into a unified virtual space, coordinated by several anchor nodes. In the deformation process, we transfer components of nodes therein into affinitive anchors by learning their correlations, and build a multi-granularity feature space calibrated with anchors. Anisotropic convolutional kernels can be further performed over the anchor-coordinated space to well encode local variations of receptive fields. By parameterizing anchors and stacking coarsening layers, we build a graph deformer network in an end-to-end fashion. Theoretical analysis indicates its connection to previous work and shows the promising property of graph isomorphism testing. Extensive experiments on widely-used datasets validate the effectiveness of GDN in graph and node classifications.

On Equienergetic, Hyperenergetic and Hypoenergetic Graphs

The eigenvalue of a graph G is the eigenvalue of its adjacency matrix and the energy E(G) is the sum of absolute values of eigenvalues of graph G. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if E(G1) = E(G2). The graphs whose energy is greater than that of complete graph are called hyperenergetic and the graphs whose energy is less than that of its order are called hypoenergetic graphs. The natural question arises: Are there any pairs of equienergetic graphs which are also hyperenergetic (hypoenergetic)? We have found an affirmative answer of this question and contribute some new results.

Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.

Reconciling enumeration contradictions: Complete list of Baranov chains with up to 15 links with mathematical proof

The identification of Baranov chains is associated with the rigid subchain identification problem, which is a crucial step in several methods of structural synthesis of kinematic chains. In this paper, a systematic approach for the detection of rigid subchains, based on matroid theory, is presented and proved. Based on this approach, a novel method for the enumeration of Baranov chains is proposed. A novel algorithm is applied to a database of non-isomorphic graphs of non-fractionated zero-mobility kinematic chains. By means of the proposed algorithm, the previous results for Baranov chains presented in literature with up to 11 links are compared and validated. Furthermore, discrepancies in the number of Baranov chains with up to 13 links, presented in literature, are pointed out, discussed and the proven results are presented. Finally, the complete family of Baranov chains with up to 15 links is obtained. Examples of application of the proposed method are provided.

Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.

Multi-Channel Graph Neural Networks

The classification of graph-structured data has be-come increasingly crucial in many disciplines. It has been observed that the implicit or explicit hierarchical community structures preserved in real-world graphs could be useful for downstream classification applications. A straightforward way to leverage the hierarchical structure is to make use the pooling algorithms to cluster nodes into fixed groups, and shrink the input graph layer by layer to learn the pooled graphs.However, the pool shrinking discards the graph details to make it hard to distinguish two non-isomorphic graphs, and the fixed clustering ignores the inherent multiple characteristics of nodes. To compensate the shrinking loss and learn the various nodes’ characteristics, we propose the multi-channel graph neural networks (MuchGNN). Motivated by the underlying mechanisms developed in convolutional neural networks, we define the tailored graph convolutions to learn a series of graph channels at each layer, and shrink the graphs hierarchically to en-code the pooled structures. Experimental results on real-world datasets demonstrate the superiority of MuchGNN over the state-of-the-art methods.

Quantifiers, generalized

Generalized quantifiers are logical tools with a wide range of uses. As the term indicates, they generalize the ordinary universal and existential quantifiers from first-order logic, ‘∀x’ and ‘∃x’, which apply to a formula A(x), binding its free occurrences of x. ∀xA(x) says that A(x) holds for all objects in the universe and ∃xA(x) says that A(x) holds for some objects in the universe, that is, in each case, that a certain condition on A(x) is satisfied. It is natural then to consider other conditions, such as ‘for at least five’, ‘at most ten’, ‘infinitely many’ and ‘most’. So a quantifier Q stands for a condition on A(x), or, more precisely, for a property of the set denoted by that formula, such as the property of being non-empty, being infinite, or containing more than half of the elements of the universe. The addition of such quantifiers to a logical language may increase its expressive power. A further generalization allows Q to apply to more than one formula, so that, for example, Qx(A(x),B(x)) states that a relation holds between the sets denoted by A(x) and B(x), say, the relation of having the same number of elements, or of having a non-empty intersection. One also considers quantifiers binding more than one variable in a formula. Qxy,zu(R(x,y),S(z,u)) could express, for example, that the relation (denoted by) R(x,y) contains twice as many pairs as S(z,u), or that R(x,y) and S(z,u) are isomorphic graphs. In general, then, a quantifier (the attribute ‘generalized’ is often dropped) is syntactically a variable-binding operator, which stands semantically for a relation between relations (on individuals), that is, a second-order relation. Quantifiers are studied in mathematical logic, and have also been applied in other areas, notably in the semantics of natural languages. This entry first presents some of the main logical facts about generalized quantifiers, and then explains their application to semantics.